We study the hydrodynamics of superfluid neutron stars, focusing on the nature of the oscillation spectrum, the effect of mutual friction force on the oscillations and the spin-up phase of pulsar glitches. We linearize the dynamical equations of a Newtonian two-fluid model for rapidly rotating backgrounds. In the axisymmetric equilibrium configurations, the two-fluid components corotate and are in β-equilibrium. We use analytical equations of state that generate stratified and non-stratified stellar models, which enable us to study the coupling between the dynamical degrees of freedom of the system. By means of time-evolutions of the linearized dynamical equations, we determine the spectrum of axisymmetric and non-axisymmetric oscillation modes, accounting for the contribution of the gravitational potential perturbations, that is, without adopting the Cowling approximation. We study the mutual friction damping of the superfluid oscillations and consider the effects of the non-dissipative part of the mutual friction force on the mode frequencies. We also provide technical details and relevant tests for the hydrodynamical model of pulsar glitches discussed by Sidery, Passamonti & Andersson. In particular, we describe the method used to generate the initial data that mimic the pre-glitch state and derive the equations that are used to extract the gravitational-wave signal.